Adopting a systematic procedure, it was seen that the incorporation of symmetries requires the addition of adequate terms to the Lagrangian density. Already in the case of the complex scalar field, there is a global gauge symmetry, which becomes local when coupled to the electromagnetic field. In this context, gauge symmetries can be viewed as rotations in the internal space parametrized by the real and imaginary parts of the field [1]}. For the vacuum electromagnetic field, we have a local gauge symmetry, which is in accordance with the causality principle. The accompanying linearity of the equations, is associated with a Noether conserved current not always emphasized in the literature. It was the main purpose of the present study, to pursue the calculations of Noether symmetries without ab initio assumptions. Such a procedure was shown to be useful for the clear identification of the internal symmetries due to linearity, in the case of the free theories (real and complex scalar field, vacuum Maxwell's equations). The conservation laws can be used to check the accuracy of numerical methods. It is expected, that the present systematic procedure could be more frequently applied, in both relativistic and non-relativistic studies. An example was applied, for the first time, to the case of a charged scalar particle under a general external electromagnetic field. The Noether symmetry condition, in this case, was reduced to the compact system provided by (REF ), which is manifestly gauge-invariant. ¨
